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Solve the differential equation (d-1)²y = 6e^x for y.

A) y = 6e^x + C_1e^x + C_2e^{2x}
B) y = 6e^x + C_1e^x + C_2e^{-x}
C) y = 6e^x + C_1e^{-x} + C_2e^{-2x}
D) y = 6e^x + C_1e^{-x} + C_2e^{x}

User Inese
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Final answer:

The differential equation (d-1)²y = 6e⁻ᴇ is solved by determining the characteristic equation and its roots, leading to the general solution involving repeated roots. The answer is D) y = 6e⁻ᴇ + C1e⁻ᴇ + C2xe⁻ᴇ.

Step-by-step explanation:

To solve the differential equation (d-1)2y = 6ex, we need to recognize that this equation implies repeated linear operators acting on the function y. The repeated operator (d - 1) suggests that we are dealing with repeated roots of the characteristic equation associated with a linear differential equation with constant coefficients. Let's proceed step by step.

First, we identify the characteristic equation corresponding to (d - 1)2, which is (r - 1)2 = 0. The roots are r = 1, both roots being the same. Thus, the homogeneous part of the solution will involve terms like C1ex and C2xex because of the repeated root.

To find a particular solution to the inhomogeneous equation, we can try a solution of the form Aex since the right-hand side is 6ex. Plugging this form into the differential equation gives us A = 6.

Therefore, the general solution to the differential equation is y = 6ex + C1ex + C2xex. Rearranging terms, we get y = (6 + C1)ex + C2xex. This implies that the correct answer is D) y = 6ex + C1ex + C2xex, since other variations with negative exponents or different powers of ex do not match the form we derived.

User Emeka Obianom
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